Hydroponic system evaluation in urban farming via fuzzy EDAS and TODIM methods

Abstract

Increasing population in the world drives people to find a different type of feeding regime. Even if there is an immense augmentation in crowd brilliant innovators are looking for new ways of farming more efficiently. Hydroponics is one of the novel paths that is a planting system without soil. The system reduces water usage by 95% and with the same rate provides efficiency in the crop, furthermore, sustainability is highly supplied. Traditional smart farming applied in the rural area strains immense transportation and brokership costs. In these days innovators make smart agriculture in vessel containers. Especially vertical and smart farming made in the suburban area of the cities offers new opportunities on vegetables’ abundance. In this paper, the efficiency of this offered system is examined with minimizing the investment cost data. The system itself and the investment area have abounded with myriad uncertainties. Fuzzy logic tackles with those vaguenesses and fuzzy Evaluation Based on Distance from Average Solution (EDAS) method supplies assistance in the decision-making process of system evaluation. In addition, TODIM (a risk sensitive iterative multi-criteria decision making method based on Prospect Theory) is employed to check the evaluation of those three alternatives and to monitor how risk perception affects decision processes. A micro-based application is performed and attractive results are achieved.

Keywords

Vertical urban agriculture fuzzy EDAS method fuzzy TODIM investment cost smart farming

1 Introduction

Vertical farming is a form of urban agriculture. According to Vertical Farm Institute, growing vegetables allows us for every one square meter of vertical farming to produce approximately the same amount of crops as 50 square meters of traditionally worked farm land.

Hydroponic system is a layout for an inexpensive and practical system that reduces water waste, and is a constituent of vertical farming. The future of hydroponic system is looking bright thanks to its nature that could be integrated with automation of nutrient and water supply. The studies conducted in this area are based on developing the strength of nutrient solution in order to improve production volume [10]. To optimize long-term plant production plan in a hydroponic system, intelligent control systems based on genetic algorithms and neural networks are used appropriately by Morimoto et al. [8].

Yuan et al. [15] examined the vertical farming effect on ventilation performance at tropical cities. Zhang et al. [16] investigated a feasibility analysis of launching vertical agriculture on university campuses. Vertical Smart Farming is a practice of vertically food production with use of sensors, detectors and obviously Internet of Things (IoT). IoT in agricultural and farming applications are made with sensors that provide intelligent and smart services towards smart agriculture. On going innovations on IoT and Big Data achieve better farming outcomes. Xin and Zazueta utilized data integration, farm management, knowledge based software as new challenges presented in smart farming [14]. Huh proposed the lightweight intrusion detection system (IDS) for the data security in vertical smart farming [7].

In this study, three alternatives called Container, Modular, and industrial site warehouse systems will be evaluated for vertical farming by urban area, with fuzzy EDAS and TODIM (an acronym in Portuguese-TOmada de Decis $\tilde{a}$ o Iterativa Multicritério) techniques. In all systems hydroponic agriculture will be utilized. In the assessment procedure of those alternatives, evaluating in one dimension orientates us in the wrong way. Naturally, this problem has to be dealt in a multi dimensional way, through multi-criteria decision making (MCDM) methods fit very well for this solution process. In addition, the topic is fresh and obtaining cost data is tough, so, fuzzy logic capable of calculation with vagueness, has integrated to the EDAS and TODIM methods-two most utilized MCDM techniques. The cited techniques are attaining popularity in recent years due to their practicality, and in EDAS side distance to averages providing better performance in the solution, then in TODIM side risk sensitive solution is obtained. Ecer [2] analyzed a hybrid model with fuzzy EDAS in the selection procedure of third-party logistics provider. Ghorabee et al. [5] extended EDAS with interval type-2 fuzzy sets and applied to supplier evaluation with environmental criteria. The motivation of that paper is to establish a decision support for the selection among three alternatives by fuzzy EDAS method including investment costs [4]. TODIM is an MCDM method based on prospect theory and already has been applied in different areas. Tosun and Akyuz [12] used fuzzy TODIM method for supplier selection. Qin et al. [9] used TODIM method based on interval Type-2 fuzzy numbers for green supplier selection. Zhang et al. [17] employed TODIM for selection of a venture capital project. Wang et al. [13] proposed an approach based on TODIM for logistics outsourcing. Duarte [1] evaluated six Brazilian Banks valuation with TODIM. There are many studies in literature merging TODIM with other MCDM methods such as Tolga’s [11] work on ERP software evaluation with AHP integrated TODIM. This paper aims to contribute by employing MCDM methods to a relatively fresh topic. We also expect this paper to be used as a learning and teaching material. Therefore EDAS and TODIM methods extended with classical fuzzy sets are employed for this study. We hope future studies will employ different extensions on the very subject soon. In this study, the following section introduces criteria for the selection of the vertical farming alternatives. EDAS method extended with trapezoidal fuzzy numbers is presented in the third section. The fourth section provides the TODIM method extended with trapezoidal fuzzy numbers. A real case application with those two trending methods, and their outcomes are delivered in the fifth section. The last section exhibits the concluding remarks and future study offerings.

2 Evaluation criteria

In the assessment process of hydroponic alternatives, at first, attributes for valuation have to be determined. The literature was searched, and structured interviews were held with people from the edge of the city who were engaged in vertical agriculture. The criteria below were detected for the evaluation process: C1) Number of Crop Types: In principle, most crops can be grown in a vertical farming. Generally, current production involves lettuce species, cherry tomatoes (less than 2.5 cm) and strawberries. Other crops, including tree fruits and salad tomatoes (5-7.5 cm) are also feasible options. But these may require more gap and space. It is quite important for the farmer to be able to respond market’s demand. Since crop prices are volatile through out the year, capability to produce different crop types is an asset for the farmer. Since hydroponics enable short maturation times, farmer may harvest most valuated crop according to the season. C2) Production Volume: In terms of production volume, it needs to be noted that greater production volumes can be achieved by more construction plants around the city. On the other hand the increasing number of plants may create a need for a complicated organizational structure. Greater volumes of production may enable the producer to increase profitability and market share. However marketing capabilities are also important for the produced crops to be sold while fresh. C3) Venture Capital Attractiveness: Indoor farming, especially vertical farming, is a promising business for investment from government, investors and universities. But this keen interest depends on the type of vertical farming. Subject is gaining popularity around the world and since the subject is still fresh more start ups enter the market. Multi national organizations, such as European Union (EU), also support agricultural projects. C4) Sustainability: There is no need for farm machinery such as tractors, trucks and harvesters which increase carbon footprint. However, the location of food production affects directly carbon footprint based on transportation. Hydroponics use less water and since the production process is soilless there is very limited need for chemicals to be used as pesticides. Overall system does not produce any known harm to the nature. C5) Flexibility (Modularity and Scalability): The modularity feature of a structure provides ability to be adaptable to workplace everywhere. And the dimension of the structure could be changed in case of need. Producer may scale the production area and production volume accordingly. C6) Workforce Requirement: Even if some alternatives provide smaller areas for planting, whole of them needs workforce for plant collecting at least. Since the proposed systems are more complicated than conventional food producing methods, educated workforce has the utmost importance. This situation increases the workforce costs and training time. C7) Stock-Out Cost: If a customer is unwilling to wait for their order to be fulfilled then they could back order the item which may cause client loss. This means that the vendor will incur some costs due to the stock-out. C8) Transportation Costs: The location of food production in closer proximity to consumers in cities is an interesting notion and results in reduction in transportation costs. Transportation vehicles also becomes smaller to be utilized in urban areas. C9) Investment Costs: In many cities, such as Istanbul and Ankara, start-up costs relate mainly to very expensive real estate in urban areas compared with rural land also depreciation costs on plant and machinery. A brief summary of criteria can be seen on Table 1.

Table 1
Evaluation Criteria for Vertical Agriculture

Criterion Description Benefit/Cost

C₁: Number of Crop Types Ability to produce different kinds of crops Benefit

C₂: Production Volume Each alternative’s capacity for maximum level crop output from a single production cycle Benefit

C₃: Venture Capital Attractiveness This criterion refers to possible investment opportunities Benefit

C₄: Sustainability Environment-friendliness of production and delivery process Benefit

C₅: Flexibility Capability to increase and decrease production volume and set-up production system at different dimensions Benefit

C₆: Workforce Requirement Workforce requirement in terms of needed personnel and training Cost

C₇: Stock-Out Cost Inability to satisfy the demand Cost

C₈: Transportation Costs Transportation needs to deliver the crops to clients Cost

C₉: Investment Costs System and facility costs Cost

Criterion	Description	Benefit/Cost
C₁: Number of Crop Types	Ability to produce different kinds of crops	Benefit
C₂: Production Volume	Each alternative’s capacity for maximum level crop output from a single production cycle	Benefit
C₃: Venture Capital Attractiveness	This criterion refers to possible investment opportunities	Benefit
C₄: Sustainability	Environment-friendliness of production and delivery process	Benefit
C₅: Flexibility	Capability to increase and decrease production volume and set-up production system at different dimensions	Benefit
C₆: Workforce Requirement	Workforce requirement in terms of needed personnel and training	Cost
C₇: Stock-Out Cost	Inability to satisfy the demand	Cost
C₈: Transportation Costs	Transportation needs to deliver the crops to clients	Cost
C₉: Investment Costs	System and facility costs	Cost

EDAS method extended with trapezoidal fuzzy numbers is presented in the next section.

3 Fuzzy EDAS method

Vertical smart farming is a new trend topic in sustainability. Through its novelty, precise data cannot be reached. There are many differences in the data obtained. This vagueness strains us to utilize fuzzy logic in our study. The dynamic structure of the vertical smart farming evaluation problem has many dimensions. This multi-dimensionality problem can be solved with multi-criteria decision making methods easily. Here, through its stability we preferred EDAS method extended with fuzzy logic.

Although fuzzy logic is known since 1965 and scientists are very familiar with the topic; due to a different defuzzification method is utilized in that study, brief information on fuzzy sets and defuzzification has to be given owing to the stream of the subject:

In this study, through its practicality and less complexity, we will utilize type-1 fuzzy sets with trapezoidal membership function. A trapezoidal fuzzy number can be denoted as $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ , a₁ ≤ a₂ ≤ a₃ ≤ a₄. Its membership function $μ_{\tilde{A}} (x) : R \to [0, 1]$ is equal to:

$μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - a_{1}}{a_{2} - a_{1}} & , & x \in [a_{1}, a_{2}] \\ 1 & , & x \in [a_{2}, a_{3}] \\ \frac{a_{4} - x}{a_{4} - a_{3}} & , & x \in [a_{3}, a_{4}] \\ 0 & , & others \end{matrix}$ (1)

The arithmetic operations between trapezoidal fuzzy numbers could be investigated from the literature easily, ergo they are not given here in order not to write the same things over and over again. The distance between any two fuzzy numbers (called $\tilde{A}$ and $\tilde{B}$ ) is calculated by the following equation:

$d (\tilde{A}, \tilde{B}) = \sqrt{\frac{[(a_{1} - b_{1})^{2} + 2 (a_{2} - b_{2})^{2} + 2 (a_{3} - b_{3})^{2} + (a_{4} - b_{4})^{2}]}{6}}$ (2)

For the defuzzified value (κ) of any fuzzy number and its maximum distance between zero (ψ) is applied as follows [3]: $κ (\tilde{A}) = \frac{1}{3} (a_{1} + a_{2} + a_{3} + a_{4} - \frac{a_{3} a_{4} - a_{1} a_{2}}{(a_{3} + a_{4}) - (a_{1} + a_{2})})$ (3)

$ψ (\tilde{A}) = {\begin{matrix} \tilde{A} & if κ (\tilde{A}) > 0 \\ \tilde{0} & if κ (\tilde{A}) \leq 0 \end{matrix}$ (4) where $\tilde{0} = (0, 0, 0, 0)$ .

EDAS was first developed by Ghorabee et al. and they had applied the method to an inventory management problem. Just a year later Ghorabee et al. [3] extended the crisp EDAS method to fuzzy logic. They tried to find a solution with the mentioned method to the supplier selection problem. The same procedure in that study will be implemented in our study as below.

Suppose there are k decision-makers (D = {D₁, D₂, . . . , D_k}) and they have n alternatives (A = {A₁, A₂, . . . , A_n}) and m criteria (C = {C₁, C₂, . . . , C_m}). Extended fuzzy EDAS method’s steps are indicated as follows:

The steps of the applied approach are as follows:

Step 1: Establish the decision matrix DM_z of the zth decision maker as shown below (gather the data from Table 2 by surveys or interviews):

Table 2

Importance terms in case of linguistic terms for criteria

Linguistic Terms	Fuzzy Number
Lowest (LT)	(0.0, 0.0, 0.0, 0.1)
Very low (VL)	(0.0, 0.1, 0.2, 0.3)
Low (L)	(0.2, 0.3, 0.4, 0.5)
Medium (M)	(0.4, 0.5, 0.5, 0.6)
High (H)	(0.5, 0.6, 0.7, 0.8)
Very High (VH)	(0.7, 0.8, 0.9, 1.0)
Highest (HT)	(0.9, 1.0, 1.0, 1.0)

${DM}_{z} = {[\begin{matrix} {\tilde{P}}_{ij}^{z} \end{matrix}]}_{n \times m} = [\begin{matrix} {\tilde{P}}_{11}^{z} & {\tilde{P}}_{12}^{z} & \dots & {\tilde{P}}_{1 m}^{z} \\ {\tilde{P}}_{21}^{z} & {\tilde{P}}_{22}^{z} & \dots & {\tilde{P}}_{2 m}^{z} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ {\tilde{P}}_{n 1}^{z} & {\tilde{P}}_{n 2}^{z} & \dots & {\tilde{P}}_{nm}^{z} \end{matrix}]$ (5) where ${\tilde{P}}_{ij}^{z}$ denotes the performance value of alternative a_i on criterion c_j assigned by the zth decision maker, 1 ≤ i ≤ n, 1 ≤ j ≤ m, and 1 ≤ z ≤ k.

Step 2: Establish the average decision matrix $\bar{G}$ as shown below:

${\tilde{P}}_{ij} = (({\tilde{P}}_{ij}^{1} \oplus {\tilde{P}}_{ij}^{2} \oplus \dots \oplus {\tilde{P}}_{ij}^{k}) / k)$ (6) $\bar{G} = {[\begin{matrix} {\tilde{P}}_{ij} \end{matrix}]}_{n \times m}$ (7) where ${\tilde{P}}_{ij}$ denotes the average performance value of alternative a_i on criterion c_j, 1 ≤ i ≤ n, 1 ≤ j ≤ m.

Step 3: Establish the weighting matrix W_z of the criteria of the zth decision maker (by applying the linguistic terms in Table 3), as shown below:

Table 3

Preference ratings in case of linguistic terms for alternatives

Linguistic Terms	Fuzzy Number
Very Poor (VP)	(0, 0, 0, 1)
Poor (P)	(0, 1, 2, 3)
Medium Poor (MP)	(2, 3, 4, 5)
Medium/Fair (M)	(4, 5, 5, 6)
Medium Good (MG)	(5, 6, 7, 8)
Good (G)	(7, 8, 9, 10)
Very Good (VG)	(9, 10, 10, 10)

$W_{z} = {[\begin{matrix} {\tilde{w}}_{j}^{z} \end{matrix}]}_{m \times 1} = [\begin{matrix} {\tilde{w}}_{1}^{z} \\ {\tilde{w}}_{2}^{z} \\ ⋮ \\ {\tilde{w}}_{m}^{z} \end{matrix}]$ (8) where ${\tilde{w}}_{j}^{z}$ denotes the weight of criterion c_j assigned by the zth decision maker, 1 ≤ j ≤ m, 1 ≤ z ≤ k.

Step 4: Establish the average weighting matrix $\bar{W}$ as shown below:

${\tilde{w}}_{j} = (({\tilde{w}}_{j}^{1} \oplus {\tilde{w}}_{j}^{2} \oplus \dots \oplus {\tilde{w}}_{j}^{k}) / k)$ (9) $\bar{W} = {[\begin{matrix} {\tilde{w}}_{j} \end{matrix}]}_{1 \times m}$ (10)

Step 5: Establish the matrix of average solutions, AV:

$AV = [{\tilde{av}}_{j}]_{1 \times m}$ (11) ${\tilde{av}}_{j} = \frac{1}{n} \sum_{i = 1}^{n} {\tilde{P}}_{ij}$ (12)

here ${\tilde{av}}_{j}$ denotes the average solutions with respect to each criterion.

Step 6: Assume B is designated for the beneficial criteria set and N for the non-beneficial criteria set. Calculate the positive distance from average (PDA) and the negative distance from average (NDA) matrices considering the type of criteria (B or N) as shown below:

$PDA = [{\tilde{pda}}_{ij}]_{n \times m}$ (13) $NDA = [{\tilde{nda}}_{ij}]_{n \times m}$ (14) ${\tilde{pda}}_{ij} = {\begin{matrix} \frac{ψ ({\tilde{P}}_{ij} ⊖ {\tilde{av}}_{j})}{κ ({\tilde{av}}_{j})} & if j \in B \\ \frac{ψ ({\tilde{av}}_{j} ⊖ {\tilde{P}}_{ij})}{κ ({\tilde{av}}_{j})} & if j \in N \end{matrix}$ (15) ${\tilde{nda}}_{ij} = {\begin{matrix} \frac{ψ ({\tilde{av}}_{j} ⊖ {\tilde{P}}_{ij})}{κ ({\tilde{av}}_{j})} & if j \in B \\ \frac{ψ ({\tilde{P}}_{ij} ⊖ {\tilde{av}}_{j})}{κ ({\tilde{av}}_{j})} & if j \in N \end{matrix}$ (16) here, ${\tilde{pda}}_{ij}$ and ${\tilde{nda}}_{ij}$ indicate the performance value of ith alternative from the average solution in terms of jth criterion’s positive and negative distances, respectively.

Step 7: Compute the positive and negative distances’ weighted sum for whole alternatives, as shown below:

${\tilde{ps}}_{i} = \sum_{j = 1}^{m} ({\tilde{w}}_{j} \otimes {\tilde{pda}}_{ij})$ (17) ${\tilde{ns}}_{i} = \sum_{j = 1}^{m} ({\tilde{w}}_{j} \otimes {\tilde{nda}}_{ij})$ (18)

Step 8: Compute the normalized values of ${\tilde{ps}}_{i}$ and ${\tilde{ns}}_{i}$ for all alternatives as below:

${\tilde{nps}}_{i} = \frac{{\tilde{ps}}_{i}}{{max}_{i} (κ ({\tilde{ps}}_{i}))}$ (19) ${\tilde{nns}}_{i} = 1 - \frac{{\tilde{ns}}_{i}}{{max}_{i} (κ ({\tilde{ns}}_{i}))}$ (20)

Step 9: Compute the assessment score ( ${\tilde{as}}_{i}$ ) for all alternatives, as shown below:

${\tilde{as}}_{i} = \frac{({\tilde{nps}}_{i} \oplus {\tilde{nns}}_{i})}{2}$ (21)

Step 10: Rank the alternatives considering the decreasing assessment scores according to $κ ({\tilde{as}}_{i})$ . Or to put it another way, the alternative which has the highest assessment score is the best choice among the nominee alternatives.

TODIM method extended with trapezoidal fuzzy numbers is presented in the next section.

4 Fuzzy TODIM method

TODIM is an MCDM method based on Nobel economic prize winning Prospect Theory which asserts that people are more prone to take risks in case of loss and more reluctant to take risk in case of gain. TODIM is first proposed by Gomes and Lima [6] based on crisp numbers. Fuzzy TODIM is a method based on fuzzy numbers. TODIM method assesses each alternative to one another for each criterion according to gains and losses.

To be more understandable; and in order not to break the rhetoric, all the notations in extended EDAS method will be used in the same way. And the first two steps of the methods are analogous although the rest of the steps are disparate.

Step 1: Constitute decision makers’ matrix according to the survey results as in Eq. 5 (gather the data from Table 2 by surveying or interviews, except Cost criterion) separately. There ${\tilde{P}}_{ij}^{z}$ denotes the performance value of alternative a_i on criterion c_j assigned by the zth decision maker, 1 ≤ i ≤ n, 1 ≤ j ≤ m, and 1 ≤ z ≤ k.

Step 2: Agglomerate all the decision makers’ opinions to form the average decision matrix, $\bar{G}$ , as the production of Eq.s 6, and 7. There denotes the average performance value of alternative a_i on criterion c_j, 1 ≤ i ≤ n, 1 ≤ j ≤ m.

Step 3: Construct the average weight matrix by applying the expert’s opinions via from the Table 3 and compute the weight matrix by Eq. 10.

Step 4: Compute the normalized ${\bar{G}}^{ν}$ matrix with maximum normalisation in beneficial criteria and minimum normalization in cost criteria as seen in Eq. below respectively:

${\bar{G}}^{ν} = {[\begin{matrix} \underset{ij}{{\tilde{P}}^{ν}} \end{matrix}]}_{n \times m} = {\begin{matrix} n \times m for beneficial criteria \\ [\tilde{P} \underset{\min}{ij cj} ø {\tilde{P}}_{ij}]_{n \times m} for non - beneficial criteria \end{matrix}$ (22)

Apply this step to whole criteria, c_j, and here 1 ≤ j ≤ m.

Step 5: Calculate the relative criteria weights ${\tilde{w}}_{{rc}_{j}}$ by normalizing the average weights derived from experts’ viewpoint as in Eq. below:

${[\begin{matrix} {\tilde{w}}_{{rc}_{j}} \end{matrix}]}_{1 \times m} = {[\begin{matrix} {\tilde{w}}_{c_{j}} ø {\tilde{w}}_{r} \end{matrix}]}_{1 \times m}$ (23) where ${\tilde{w}}_{r}$ is the reference criterion weight with the highest value accorded to its importance. And ${\tilde{w}}_{c_{j}}$ is denoted as every criterion’s weights, and 1 ≤ j ≤ m.

Step 6: Obtain the contribution of any criterion c_j to the dominance of any alternative A_{a
_i} over each alternative A_i; ${\tilde{ϕ}}_{c_{j}} (A_{a_{i}}, A_{i})$ , by Eq. below:

${\tilde{ϕ}}_{c_{j}} (A_{a_{i}}, A_{i}) = {\begin{matrix} \sqrt{\frac{{\tilde{w}}_{{rc}_{j}}}{\sum_{j = 1}^{m} {\tilde{w}}_{{rc}_{j}}} d ({\tilde{P}}_{a_{i} j}^{v}, {\tilde{P}}_{ij}^{v})}, & if κ ({\tilde{P}}_{a_{i} j}^{v}) - κ ({\tilde{P}}_{ij}^{v}) > 0 \\ 0, & if κ ({\tilde{P}}_{a_{i} j}^{v}) - κ ({\tilde{P}}_{ij}^{v}) = 0 \\ - \frac{1}{θ} \sqrt{\frac{\sum_{j = 1}^{m} {\tilde{w}}_{{rc}_{j}}}{{\tilde{w}}_{{rc}_{j}}} d ({\tilde{P}}_{a_{i} j}^{v}, {\tilde{P}}_{ij}^{v})}, & if κ ({\tilde{P}}_{a_{i} j}^{v}) - κ ({\tilde{P}}_{ij}^{v}) < 0 \end{matrix}$ (24) where the term (A_{a
_i}, A_i) presents the referred alternative over other i alternatives, $d ({\tilde{P}}_{a_{i} j}^{v}, {\tilde{P}}_{ij}^{v})$ exhibits the distance between performance values of each criterion j to each alternative i. $κ ({\tilde{P}}_{a_{i} j}^{v})$ , and $κ ({\tilde{P}}_{ij}^{v})$ represent the type reduced crisp values of related performance values, respectively and they are deduced by Eq. 3.

Step 7: Compute the dominance of each alternative over each other alternatives by using the Eq. below:

$\begin{matrix} \tilde{δ} (A_{a_{i}}, A_{i}) = \sum_{j = 1}^{m} {\tilde{ϕ}}_{c_{j}} (A_{a_{i}}, A_{i}) & \forall (a_{i}, c_{j}) \end{matrix}$ (25)

where 1 ≤ i ≤ n, and 1 ≤ j ≤ m.

Step 8: Transpose the dominance values and constitute the ${[\begin{matrix} \tilde{δ} (A_{i}, A_{a_{i}}) \end{matrix}]}_{n \times n}$ matrix for ease of computation.

Step 9: Calculate the normalized global value of alternative i by using the equation below:

${\tilde{ξ}}_{i} = \frac{\sum \tilde{δ} (A_{a_{i}}, A_{i}) - \min \sum \tilde{δ} (A_{a_{i}}, A_{i})}{\max \sum \tilde{δ} (A_{a_{i}}, A_{i}) - \min \sum \tilde{δ} (A_{a_{i}}, A_{i})}$ (26)

where $\min \sum \tilde{δ} (A_{a_{i}}, A_{i})$ represents the minimum of sum of dominance value of each alternative and $\max \sum \tilde{δ} (A_{a_{i}}, A_{i})$ represents the maximum of sum of dominance value of each alternative also. Here; 1 ≤ i ≤ n.

Step 10: Attain the type-reduced global values of each alternative i, $κ ({\tilde{ξ}}_{i})$ by Eq. 3, then rank the alternatives in descending sequence to find the best one or the accurate ordering.

A real case application with Fuzzy EDAS and Fuzzy TODIM methods, and their outcomes are delivered in the next section.

5 Application

In this study, the multi-criteria decision making methods expressed in the preceding section are performed to select the best alternative among vertical farming projects in a city region. At first we analyzed the problem with fuzzy EDAS method. Then, due to our investigation on the said problem is in an unattempted area, it has to be checked with an another risk sensitive method called fuzzy TODIM. At this section all the solution procedures for those two MCDM methods are given.

The first alternative is a system in a Container, which has LEDs, six vertical farming slots, water tank, air conditioner, refrigerator, other necessary equipment, and 21-meter square area. The second alternative is a novel Modular system, which has similar equipment in the former alternative with lower cost due to its installation flexibility. The last alternative is a system in an industrial site warehouse, which has more equipment, and 200-meter square area. The criteria are obtained from the literature and inquired to the experts. Five experts (two of them are an agricultural engineer, and three of them are a farmer) dealing with hydroponic farming are asked for a survey to evaluate the criteria and alternatives. One of the farmers located in Buyukcekmece region, who had worked in the production of tissue (seedling), also established a product development laboratory. Another farmer/expert grows strawberries by the hydroponic farming in the same region of Istanbul in a greenhouse. A consultant/expert has made the hydroponic farming, however now he markets the system and advises others. The agricultural engineers observe those farmers engaging said farming systems, and advise to them.

The cost data is acquired from the market and determined according to the designs of each system separately. Container alternative is 21 m² area, however industrial site alternative has 200 m² area. And in addition; to compare the alternatives accurately, we took modular alternative’s area just like in container alternative. Whole data given in Table 4 is integrated to the solution techniques in their Step 2 as a non-benefit criterion. Even though cost estimates for container and modular alternatives are pretty close, two systems differentiate on other aspects regarding our other criteria. Decision makers’ perception of these two alternatives also different as can be seen in Table 5. Those data are costs for a unit square meter of each alternative.

Table 4
Cost Estimations of Vertical Farming System Alternatives

Cost Item Container () Modular () Ind. Site W. ()

LED (150 m / 1170 m)) (2533.2, 2831.2, 3129.2, 3427.2) (2533.2, 2831.2, 3129.2, 3427.2) (98793.6, 110416.4, 122039.2, 133662.0)

Water hose (40 m / 240 m) (467.5, 522.5, 577.5, 632.5) (467.5, 522.5, 577.5, 632.5) (2805.0, 3135.0, 3465.0, 3795,0)

Water hose elbow (6 pieces) (76.5, 85.5, 94.5, 103.5) (76.5, 85.5, 94.5, 103,5)

Capillary water hose (30 m) (80.1, 89.5, 98.9, 108.3) (80.1, 89.5, 98.9, 108.3)

Suspension rope (30 m) (34.0, 38.0, 42.0, 46.0) (34.0, 38.0, 42.0, 46.0)

Hook (300 pieces) (453.9, 507.3, 560.7, 614.1) (453.9, 507.3, 560.7, 614.1)

Plastic Velcro Clamp (100 pieces) (4.3, 4.8, 5.4, 5.9) (4.3, 4.8, 5.4, 5.9)

PVC (150 m) (3187.5, 3562.5, 3937.5, 4312.5) (3187.5, 3562.5, 3937.5, 4312.5)

Water tank (500 / 2000 liters) (425.0, 475.0, 525.0, 575.0) (425.0, 475.0, 525.0, 575.0) (1700.0, 1900.0, 2100.0, 2300.0)

Shutter / Container (21 m²) (8500.0, 9500.0, 10500.0, 11500.0) (6375.0, 7125.0, 7875.0, 8625.0)

Small Water Tanks (for nutrients) (212.5, 237.5, 262.5, 287.5) (212.5, 237.5, 262.5, 287.5)

Oxygen Cylinder (467.5, 522.5, 577.5, 632.5) (467.5, 522.5, 577.5, 632.5)

Air conditioning (4250.0, 4750.0, 5250.0, 5750,0) (4250.0, 4750.0, 5250.0, 5750.0) (42500.0, 47500.0, 52500.0, 57500.0)

Electrical Installation and Equipment (1700.0, 1900.0, 2100.0, 2300.0) (1700.0, 1900.0, 2100.0, 2300.0) (4250.0, 4750.0, 5250.0, 5750.0)

Counter type refrigerator (1 / 3 pieces) (5185.0, 5795.0, 6405.0, 7015.0) (5185.0, 5795.0, 6405.0, 7015.0) (15555.0, 17385.0, 19215.0, 21045.0)

Rack (117 pieces) (64642.5, 72247.5, 79852.5, 87457.5)

Cornice (130 m) (1270.8, 1420.3, 1569.8, 1719.3)

Recirculation pump (212.5, 237.5, 262.5, 287.5)

Total (27577.0, 30821.3, 34065.7, 37310.0) (25452.0, 28446.3, 31440.7, 34435.0) (231729.4, 258991.7, 286253.9, 313516.2)

Total per 1 m² (1313.2, 1467.7, 1622.2, 1776.7) (1212.0, 1354.6, 1497.2, 1639.8) (1158.6, 1295.0, 1431.3, 1567.6)

Cost Item	Container ()	Modular ()	Ind. Site W. ()
LED (150 m / 1170 m))	(2533.2, 2831.2, 3129.2, 3427.2)	(2533.2, 2831.2, 3129.2, 3427.2)	(98793.6, 110416.4, 122039.2, 133662.0)
Water hose (40 m / 240 m)	(467.5, 522.5, 577.5, 632.5)	(467.5, 522.5, 577.5, 632.5)	(2805.0, 3135.0, 3465.0, 3795,0)
Water hose elbow (6 pieces)	(76.5, 85.5, 94.5, 103.5)	(76.5, 85.5, 94.5, 103,5)
Capillary water hose (30 m)	(80.1, 89.5, 98.9, 108.3)	(80.1, 89.5, 98.9, 108.3)
Suspension rope (30 m)	(34.0, 38.0, 42.0, 46.0)	(34.0, 38.0, 42.0, 46.0)
Hook (300 pieces)	(453.9, 507.3, 560.7, 614.1)	(453.9, 507.3, 560.7, 614.1)
Plastic Velcro Clamp (100 pieces)	(4.3, 4.8, 5.4, 5.9)	(4.3, 4.8, 5.4, 5.9)
PVC (150 m)	(3187.5, 3562.5, 3937.5, 4312.5)	(3187.5, 3562.5, 3937.5, 4312.5)
Water tank (500 / 2000 liters)	(425.0, 475.0, 525.0, 575.0)	(425.0, 475.0, 525.0, 575.0)	(1700.0, 1900.0, 2100.0, 2300.0)
Shutter / Container (21 m²)	(8500.0, 9500.0, 10500.0, 11500.0)	(6375.0, 7125.0, 7875.0, 8625.0)
Small Water Tanks (for nutrients)	(212.5, 237.5, 262.5, 287.5)	(212.5, 237.5, 262.5, 287.5)
Oxygen Cylinder	(467.5, 522.5, 577.5, 632.5)	(467.5, 522.5, 577.5, 632.5)
Air conditioning	(4250.0, 4750.0, 5250.0, 5750,0)	(4250.0, 4750.0, 5250.0, 5750.0)	(42500.0, 47500.0, 52500.0, 57500.0)
Electrical Installation and Equipment	(1700.0, 1900.0, 2100.0, 2300.0)	(1700.0, 1900.0, 2100.0, 2300.0)	(4250.0, 4750.0, 5250.0, 5750.0)
Counter type refrigerator (1 / 3 pieces)	(5185.0, 5795.0, 6405.0, 7015.0)	(5185.0, 5795.0, 6405.0, 7015.0)	(15555.0, 17385.0, 19215.0, 21045.0)
Rack (117 pieces)			(64642.5, 72247.5, 79852.5, 87457.5)
Cornice (130 m)			(1270.8, 1420.3, 1569.8, 1719.3)
Recirculation pump			(212.5, 237.5, 262.5, 287.5)
Total	(27577.0, 30821.3, 34065.7, 37310.0)	(25452.0, 28446.3, 31440.7, 34435.0)	(231729.4, 258991.7, 286253.9, 313516.2)
Total per 1 m²	(1313.2, 1467.7, 1622.2, 1776.7)	(1212.0, 1354.6, 1497.2, 1639.8)	(1158.6, 1295.0, 1431.3, 1567.6)

Table 5

Decision Makers Evaluation for each alternative

		C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇	C ₈
	A ₁	P	P	P	P	P	VG	VG	VG
D ₁	A ₂	MG	MG	MG	MG	MG	MG	MG	MG
	A ₃	VG	VG	VG	VG	VG	P	P	P
	A ₁	MP	P	MP	VG	G	VG	MP	VG
D ₂	A ₂	MG	M	G	VG	VG	G	G	MP
	A ₃	VG	VG	VG	VG	MP	M	VG	G
	A ₁	MG	MG	G	G	VG	VG	MG	VG
D ₃	A ₂	VG	VG	VG	G	VG	MG	VG	VG
	A ₃	VG	G	VG	G	G	G	G	MG
	A ₁	MP	MP	P	VG	VG	MG	MP	VG
D ₄	A ₂	MG	MG	M	M	M	MG	M	MG
	A ₃	G	VG	VG	G	MP	G	VG	M
	A ₁	G	G	G	G	G	P	VG	VG
D ₅	A ₂	MP	MP	G	G	G	P	MP	P
	A ₃	M	M	MG	MG	MG	P	MP	MP

5.1 Fuzzy EDAS application

Theoretical frame of fuzzy EDAS method was given in section 4; in this section application of that background to vertical farming alternative selection is presented.

Decision matrix has to be formed as in step 1 of the said procedure: Decision makers are kindly asked to evaluate each alternative’s performance on given criteria. Decision makers utilized qualitative ratings given in Table 3 to evaluate each alternative’s expected performance. Table 5 shows decision makers’ overall ratings for each alternative.

In Step 2, average decision matrix is established as given below: Decision makers’ overall rates need to be combined to prepare an averege decision matrix. Eq. 6 is used to calculate average values for alternatives. For example; calculation for first alternative’s performance on first criterion is indicated as:

${\tilde{P}}_{11} = ({\tilde{P}}_{11}^{1} \oplus {\tilde{P}}_{11}^{2} \oplus {\tilde{P}}_{11}^{3} \oplus {\tilde{P}}_{11}^{4} \oplus {\tilde{P}}_{11}^{5}) / 5$

${\tilde{P}}_{11} = ((0, 1, 2, 3) \oplus (2, 3, 4, 5) \oplus (5, 6, 7, 8) \oplus$ (2, 3, 4, 5) ⊕ (7, 8, 9, 10))/5

${\tilde{P}}_{11} = (3.2, 4.2, 5.2, 6.2)$

Computed performance matrix can be observed in Table 6:

Weighting matrix is constituted in step 3 by the procedure dedicated below: Decision makers are also kindly requested to utilize Table 2 to assign a value for each criterion. Decision makers’ opinions regarding criteria are showed on Table 7. Each criterion’s weight can be calculated by using Eq. 9 and average weighting matrix in step 4 can be combined by using Eq. 10. To be more understandable, ${\tilde{w}}_{1}$ is calculated as an example:

${\tilde{w}}_{1} = (((0, 0, 0, 0.1) \oplus (0.5, 0.6, 0.7, 0.8) \oplus (0.7, 0.8, 0.9, 1) \oplus . . . (0.4, 0.5, 0.5, 0.6) \oplus (0.2, 0.3, 0.4, 0.5)) / 5)$

${\tilde{w}}_{1} = (0.36, 0.44, 0.5, 0.6)$

$κ ({\tilde{w}}_{1}) = (0.36 + 0.44 + 0.5 + 0.6 - ((0.5 \times 0.6 - 0.36 \times 0.44) / . . . (0.5 + 0.6 - (0.36 + 0.44)))) / 3$

w₁ = 0.476

Table 6
Performance Matrix

A ₁ A ₂ A ₃

C ₁ (3.2, 4.2, 5.2, 6.2) (5.2, 6.2, 7, 7.8) (7.6, 8.6, 8.8, 9.2)

C ₂ (2.8, 3.8, 4.8, 5.8) (5, 6, 6.6, 7.4) (7.6, 8.6, 8.8, 9.2)

C ₃ (3.2, 4.2, 5.2, 6.2) (6.4, 7.4, 8, 8.8) (8.2, 9.2, 9.4, 9,6)

C ₄ (6.4, 7.4, 8, 8.6) (6.4, 7.4, 8, 8.8) (7.4, 8.4, 9, 9.6)

C ₅ (6.4, 7.4, 8, 8.6) (6.8, 7.8, 8.2, 8.8) (5, 6, 6.8, 7.6)

C ₆ (6.4, 7.4, 7.8, 8.2) (4.4, 5.4, 6.4, 7.4) (3.6, 4.6, 5.4, 6.4)

C ₇ (5.4, 6.4, 7, 7.6) (5.4, 6.4, 7, 7.8) (5.4, 6.4, 7, 7.6)

C ₈ (9, 10, 10, 10) (4.2, 5.2, 6, 6.8) (3.6, 4.6, 5.4, 6.4)

C ₉ (1313.2, 1467.7, (1212.0, 1354.6, (1158.65, 1294.96,

1622.2, 1776.7) 1497.2, 1639.8) 1431.27, 1567.58)

	A ₁	A ₂	A ₃
C ₁	(3.2, 4.2, 5.2, 6.2)	(5.2, 6.2, 7, 7.8)	(7.6, 8.6, 8.8, 9.2)
C ₂	(2.8, 3.8, 4.8, 5.8)	(5, 6, 6.6, 7.4)	(7.6, 8.6, 8.8, 9.2)
C ₃	(3.2, 4.2, 5.2, 6.2)	(6.4, 7.4, 8, 8.8)	(8.2, 9.2, 9.4, 9,6)
C ₄	(6.4, 7.4, 8, 8.6)	(6.4, 7.4, 8, 8.8)	(7.4, 8.4, 9, 9.6)
C ₅	(6.4, 7.4, 8, 8.6)	(6.8, 7.8, 8.2, 8.8)	(5, 6, 6.8, 7.6)
C ₆	(6.4, 7.4, 7.8, 8.2)	(4.4, 5.4, 6.4, 7.4)	(3.6, 4.6, 5.4, 6.4)
C ₇	(5.4, 6.4, 7, 7.6)	(5.4, 6.4, 7, 7.8)	(5.4, 6.4, 7, 7.6)
C ₈	(9, 10, 10, 10)	(4.2, 5.2, 6, 6.8)	(3.6, 4.6, 5.4, 6.4)
C ₉	(1313.2, 1467.7,	(1212.0, 1354.6,	(1158.65, 1294.96,
	1622.2, 1776.7)	1497.2, 1639.8)	1431.27, 1567.58)

Table 7

Decision Makers’ Opinions on Criteria importance

	D ₁	D ₂	D ₃	D ₄	D ₅
C ₁	LT	H	VH	M	L
C ₂	M	HT	H	HT	HT
C ₃	HT	HT	HT	VH	HT
C ₄	HT	HT	VH	HT	HT
C ₅	HT	HT	HT	VH	HT
C ₆	VL	HT	VH	HT	HT
C ₇	H	VH	VH	HT	HT
C ₈	M	H	VH	VH	VL
C ₉	H	H	H	VH	H

In step 5, average solutions matrix is set up as stated below: Average Solutions matrix shows average performance values regarding each criterion. A sample calculation for the first alternative is presented as:

${\tilde{av}}_{1} = (((3.2, 4.2, 5.2, 6.2) \oplus (5.2, 6.2, 7, 7.8) \oplus (7.6, 8.6, 8.8, 9.2)) / 3)$

${\tilde{av}}_{1} = (5.33, 6.33, 7, 7.73)$

Distance from averages matrices are computed as below as in step 6: Each alternative’s positive and negative distances from average solution in terms of each criterion is calculated. Calculations differ if the criterion is beneficial or not. $\tilde{nda}$ calculations are opposite of $\tilde{pda}$ calculations. The sample calculations for both beneficial and non-beneficial criteria are given below:

$\tilde{pda}$ for beneficial criterion:

${\tilde{pda}}_{31} = ((7.6, 8.6, 8.8, 9.2) ⊖ (5.33, 6.33, 7, 7.73)) / 6.59$

${\tilde{pda}}_{31} = (- 0.02, 0.24, 0.37, 0.59)$

$\tilde{pda}$ for cost criterion:

${\tilde{pda}}_{39} = ((1227.94, 1372.41, 1516.87, 1661.34)$

⊖(1158.651294.961431.271567.58))/1444.64

${\tilde{pda}}_{39} = (- 0.348 - 0.154, 0.041, 0.235)$

In step 7, distances’ weighted sums are computed according to values in the previous step. Each alternative’s both positive and negative distances’ weighted sum can be calculated with Eq. 17 and Eq. 18. Calculations below demonstrates values for third alternative.

${\tilde{w}}_{1} \otimes {\tilde{pda}}_{31} = (0.36, 0.44, 0.5, 0.6) \otimes (- 0.02, 0.24, 0.37, 0.59)$

${\tilde{w}}_{1} \otimes {\tilde{pda}}_{31} = (- 0.0073, 0.1069, 0.1872,$ 0.352185)

${\tilde{ps}}_{3} = (- 1.307, - 0.087, 0.920, 2.519)$

Distances’ weighted sums are normalized in step 8. Maximum crisp value for ${\tilde{ps}}_{i}$ that belongs to the third alternative is calculated as 0.529709. This value will be used to normalize ${\tilde{ps}}_{i}$ values of each alternative. Maximum crisp value for ${\tilde{ns}}_{i}$ that belongs to the first alternative is calculated as 0.383094. This value will be used to normalize ${\tilde{ns}}_{i}$ values of each alternative. Third alternative’s ${\tilde{nps}}_{i}$ and ${\tilde{nns}}_{i}$ calculations are shown below as examples:

${\tilde{nps}}_{3} = (- 1.307, - 0.087, 0.920, 2.519) / 0.529709$

${\tilde{nps}}_{3} = (- 2.468, - 0.164, 1.738, 4.755)$

${\tilde{nns}}_{3} = 1 - (- 0.181, 0.035, 0.225, 0.459) / 0.383094$

Assessment scores ( ${\tilde{as}}_{i}$ ) for each alternative is calculated in step 9 as seen below:

${\tilde{as}}_{1} = ((- 0.912, - 0.211, 0.392, 1.262) \oplus (- 4.652, - 0.947, 1.587, 4.212)) / 2$

${\tilde{as}}_{1} = (- 2.8415, - 0.9584, 0.3927, 1.9944)$

${\tilde{as}}_{2} = ((- 2.141, - 0.713, 0.730, 2.860) \oplus (- 2.048, 0.116, 1.534, 3.216)) / 2$

${\tilde{as}}_{2} = (- 2.094, - 0.298, 1.132, 3.038)$

${\tilde{as}}_{3} = ((- 2.468, - 0.164, 1.738, 4.755) \oplus (- 0.197, 0.413, 0.908, 1.474)) / 2$

${\tilde{as}}_{3} = (- 1.333, 0.125, 1.323, 3.114)$

At the last step of fuzzy EDAS method, in step 10, the alternatives should be ranked by defuzzification procedure given in Eq. 3. Highest score of ${\tilde{as}}_{i}$ shows the best alternative:

$κ ({\tilde{as}}_{1}) = 0.0703$ $κ ({\tilde{as}}_{2}) = 0.4496$ $κ ({\tilde{as}}_{3}) = 0, 8234$

The data obtained from the surveys and the cost data are applied in the order given in the previous section. According to fuzzy EDAS method, the leading alternative is found as the industrial site warehouse, which has more advantages in production volume and stock-out cost criteria according to others as seen from above.

5.2 Fuzzy TODIM Application

As mentioned above, first three steps of fuzzy EDAS and fuzzy TODIM methods are analogous; so the same criteria weighting and same performance rating steps of fuzzy EDAS application will be used in fuzzy TODIM application also. We will proceed with normalization of performance values. Eq. 22 will be utilized to normalize the performance values. A small sample of normalization for both beneficial and cost criteria is shown below:

For beneficial criteria:

${\tilde{P}}_{11} = (\frac{3.2}{9.2}, \frac{4.2}{8.8}, \frac{5.2}{8.6}, \frac{6.2}{7.6}) = (0.348, 0.477, 0.605, 0.816)$

For cost criteria:

${\tilde{P}}_{19} = (\frac{1776.67}{1158.65}, \frac{1622.18}{1294.96}, \frac{1467.68}{1431.27}, \frac{1313.19}{1567.58}) = (0.652, 0.798, 0.975, 1.194)$

Normalization procedure in step 4 is applied to whole criteria according to alternatives and normalized ${\bar{G}}^{ν}$ matrix is constructed as shown on Table 8:

Table 8
Normalized ${\bar{G}}^{ν}$ matrix

A ₁ A ₂ A ₃

C ₁ (0.348, 0.477, 0.605, 0.816) (0.565, 0.705, 0.814, 1.026) (0. 826, 0.977, 1.023, 1.211)

C ₂ (0.304, 0.432, 0.558, 0.763) (0.543, 0.682, 0.767, 0.974) (0.826, 0.977, 1.023, 1.211)

C ₃ (0.333, 0.447, 0.565, 0.756) (0.667, 0.787, 0.870, 1.073) (0.854, 0.979, 1.022, 1.171)

C ₄ (0.667, 0.822, 0.952, 1.162) (0.667, 0.822, 0.952, 1.189) (0.771, 0.933, 1.071, 1.297)

C ₅ (0.727, 0.902, 1.026, 1.265) (0.773, 0.951, 1.051, 1.294) (0.568, 0.732, 0.872, 1.118)

C ₆ (0.780, 0.949, 1.054, 1.281) (0.537, 0.692, 0.865, 1.156) (0.439, 0.590, 0.730, 1.000)

C ₇ (0.692, 0.914, 1.094, 1.407) (0.692, 0.914, 1.094, 1.444) (0.692, 0.914, 1.094, 1.407)

C ₈ (0.900, 1.000, 1.000, 1.111) (0.420, 0.520, 0.600, 0.756) (0.360, 0.460, 0.540, 0.711)

C ₉ (0.652, 0.798, 0.975, 1.194) (0.707, 0.865, 1.056, 1.293) (0.739, 0.905, 1.105, 1.353)

	A ₁	A ₂	A ₃
C ₁	(0.348, 0.477, 0.605, 0.816)	(0.565, 0.705, 0.814, 1.026)	(0. 826, 0.977, 1.023, 1.211)
C ₂	(0.304, 0.432, 0.558, 0.763)	(0.543, 0.682, 0.767, 0.974)	(0.826, 0.977, 1.023, 1.211)
C ₃	(0.333, 0.447, 0.565, 0.756)	(0.667, 0.787, 0.870, 1.073)	(0.854, 0.979, 1.022, 1.171)
C ₄	(0.667, 0.822, 0.952, 1.162)	(0.667, 0.822, 0.952, 1.189)	(0.771, 0.933, 1.071, 1.297)
C ₅	(0.727, 0.902, 1.026, 1.265)	(0.773, 0.951, 1.051, 1.294)	(0.568, 0.732, 0.872, 1.118)
C ₆	(0.780, 0.949, 1.054, 1.281)	(0.537, 0.692, 0.865, 1.156)	(0.439, 0.590, 0.730, 1.000)
C ₇	(0.692, 0.914, 1.094, 1.407)	(0.692, 0.914, 1.094, 1.444)	(0.692, 0.914, 1.094, 1.407)
C ₈	(0.900, 1.000, 1.000, 1.111)	(0.420, 0.520, 0.600, 0.756)	(0.360, 0.460, 0.540, 0.711)
C ₉	(0.652, 0.798, 0.975, 1.194)	(0.707, 0.865, 1.056, 1.293)	(0.739, 0.905, 1.105, 1.353)

In step 5, relative criteria weights are calculated by using experts’ opinions on criteria importance to prepare relative criteria weights. Relative criteria weights are computed by dividing normalized criteria weights to reference criterion weight from Eq. 23 as seen below: ${\tilde{w}}_{c_{1}} = (0.360, 0.440, 0.500, 0.600)$

${\tilde{w}}_{r} = (0.860, 0.960, 0.980, 1.000)$

${\tilde{w}}_{{rc}_{1}} = {\tilde{w}}_{c_{1}} ø {\tilde{w}}_{r} = (0.360, 0.449, 0.521, 0.698)$

The whole relative criteria weights are derived by the preceding formula and they are depicted in Table 9.

Table 9

Relative Criteria Weights

	${\tilde{w}}_{c_{j}}$	${\tilde{w}}_{{rc}_{j}}$
C ₁	(0.360, 0.440, 0.500, 0.600)	(0.360, 0.449, 0.521, 0.698)
C ₂	(0.720, 0.820, 0.840, 0.880)	(0.720, 0.837, 0.875, 1.023)
C ₃	(0.860, 0.960, 0.980, 1.000)	(0.860, 0.980, 1.021, 1.163)
C ₄	(0.860, 0.960, 0.980, 1.000)	(0.860, 0.980, 1.021, 1.163)
C ₅	(0.860, 0.960, 0.980, 1.000)	(0.860, 0.980, 1.021, 1.163)
C ₆	(0.680, 0.780, 0.820, 0.860)	(0.680, 0.796, 0.854, 1.000)
C ₇	(0.740, 0.840, 0.900, 0.960)	(0.740, 0.857, 0.938, 1.116)
C ₈	(0.460, 0.560, 0.640, 0.740)	(0.460, 0.571, 0.667, 0.860)
C ₉	(0.540, 0.640, 0.740, 0.840)	(0.540, 0.653, 0.771, 0.977)

Dominance matrices are prepared for each criterion in step 6. Each alternative’s dominance to one another for given criterion can be seen in this step. Dominance matrices are prepared according to Eq. 24, and for example; dominance matrix of first criterion when θ equals 1 is shown as below:

$\tilde{P_{21}} = (0.565, 0.705, 0.814, 1.026)$

$\tilde{P_{12}} = (0.348, 0.477, 0.605, 0.816)$

$\tilde{P_{21}} > \tilde{P_{12}}$

$d (\tilde{P_{21}}, \tilde{P_{12}}) = \sqrt{\frac{[{(0.565 - 0.348)}^{2} + 2 {(0.705 - 0.477)}^{2} + 2 {(0.814 - 0.605)}^{2} + {(1.026 - 0.816)}^{2}]}{6}} = 0.21698$

${\tilde{ϕ}}_{c_{1}} (A_{a_{2}}, A_{1}) = \sqrt{\frac{(0.36, 0.44897, 0.52083, 0.69767)}{(6.08, 7.10204, 7.6875, 9.16279)} * 0.21698} = (0.09233, 0.11257, 0.12614, 0.15779)$

${\tilde{ϕ}}_{c_{1}} (A_{a_{1}}, A_{2}) = - \frac{1}{1} \sqrt{\frac{(6.08, 7.10204, 7.6875, 9.16279)}{(0.36, 0.44897, 0.52083, 0.69767)} * 0.21698}$

= (-2.35006, - 1.92751, - 1.72012, - 1.37513)

${\tilde{ϕ}}_{c_{1}} = [\begin{matrix} (0, 0, 0, 0) & (0.092, 0.112, 0.126, 0.157) & (0.116, 0.142, 0.159, 0.199) \\ (- 2.350, - 1.927, - 1.720, - 1.375) & (0, 0, 0, 0) & (0.085, 0.104, 0.117, 0.146) \\ (- 3.398, - 2.787, - 2.487, - 1.988) & (- 2.458, - 2.016, - 1.799, - 1.438) & (0, 0, 0, 0) \end{matrix}]$

In step 7, overall dominance matrix is calculated by Eq. 25 as in the following example. After calculating each alternative’s dominance for each criterion, we combined these dominance values to construct overall dominance value of each alternative. Calculations for overall dominance matrix is shown below:

$\tilde{δ} (A_{a_{1}}, A_{2})$ = ${\tilde{ϕ}}_{c_{1}} (A_{a_{1}}, A_{2})$ + ${\tilde{ϕ}}_{c_{2}} (A_{a_{1}}, A_{2})$ +... + ${\tilde{ϕ}}_{c_{9}} (A_{a_{1}}, A_{2})$ = (0.092, 0.112, 0.126, 0.157) + (0.134, 0.157, 0.167, 0.196) + . . . + (0.067, 0.080, 0.091, 0.111) = (-4.053, - 3.160, - 2.732, - 1.999)

In step 8, the dominance values matrix is transposed and it is presented as below:

$Δ = [\begin{matrix} (0, 0, 0, 0) & (- 8.2, - 6.8, - 6.2, - 5.1) & (- 10.3, - 8.5, - 7.7, - 6.3) \\ (- 4.1, - 3.2, - 2.7, - 2.0) & (0, 0, 0, 0) & (- 7.3, - 6.0, - 5.5, - 4.5) \\ (- 5.1, - 4.0, - 3.5, - 2.7) & (- 3.0, - 2.4, - 2.1, - 1.6) & (0, 0, 0, 0) \end{matrix}]$

Overall values are computed for all alternatives in step 8: Each row of overall dominance matrix belongs to an alternative. Accumulated value of each row gives the overall ranking value of each alternative. First alternative’s overall ranking value is calculated as an example:

$\sum \tilde{δ} (A_{a_{1}}, A_{a_{i}}) = [(0, 0, 0, 0) \oplus (- 8.196, - 6.804, - 6.194, - 5.073) \oplus (- 10.319, - 8.524, - 7.740, - 6.290)] = (- 18.515, - 15.328, - 13.934, - 11.363)$

$\sum \tilde{δ} (A_{a_{2}}, A_{i}) = (- 11.351, - 9.188, - 8.212, - 6.458)$

$\sum \tilde{δ} (A_{a_{3}}, A_{i}) = (- 8.059, - 6.399, - 5.654, - 4.293)$

In step 9, normalized global values of each alternatives can be computed as follows:

${\tilde{ξ}}_{1} = \frac{(- 18.515, - 15.328, - 13.934, - 11.363) ⊖ (- 18.515, - 15.328, - 13.934, - 11.363)}{(- 8.059, - 6.399, - 5.654, - 4.293) ⊖ (- 18.515, - 15.328, - 13.934, - 11.363)}$ = (-0.503, -0.144, 0.185, 2.165)

${\tilde{ξ}}_{2} = \frac{(- 11.351, - 9.188, - 8.212, - 6.458) ⊖ (- 18.515, - 15.328, - 13.934, - 11.363)}{((- 8.059, - 6.399, - 5.654, - 4.293) ⊖ (- 18.515, - 15.328, - 13.934, - 11.363)}$ = (0.001, 0.491, 0.944, 3.650) ${\tilde{ξ}}_{3} = \frac{(- 8.059, - 6.399, - 5.654, - 4.293) ⊖ (- 18.515, - 15.328, - 13.934, - 11.363)}{(- 8.059, - 6.399, - 5.654, - 4.293) ⊖ (- 18.515, - 15.328, - 13.934, - 11.363)}$ = (0.232, 0.779, 1.284, 4.305)

$κ (ξ_{1}) = \frac{1}{3} (- 0.503 - 0.144 + 0.185 + 2.165 - \frac{0.185 * 2.165 - (- 0.503 * - 0.144)}{(0.185 + 2.165) - (- 0.503 - 0.144)}) = 0.5312$

$κ (ξ_{2}) = \frac{1}{3} (0.01 + 0.491 + 0.944 + 3.650 - \frac{0.944 * 3.650 - 0.001 * 0.491}{(0.944 + 3.650) - (0.001 + 0.491)}) = 1.4151$

$κ (ξ_{3}) = \frac{1}{3} (0.232 + 0.779 + 1.284 + 4.305 - \frac{1.284 * 4.305 - 0.232 * 0.779}{(1.284 + 4.305) - (0.232 + 0.779)}) = 1.8106$

A sensitivity analysis is calculated to see the effect of risk on results. θ parameter is changed to 2, 2.5, 4 and 7. Results are on Table 10:

Table 10

Sensitivity Analysis of overall values for different θ values

		κ (ξ_i)	values
Alternatives	θ = 1	θ = 2.5	θ = 3	θ = 4	θ = 7
Container	0.531	0.602	0.626	0.675	0.826
Modular	1.415	1.521	1.557	1.631	1.866
Ind. Site W.	1.811	1.959	2.010	2.114	2.441

In order to rank the alternatives as in step 10, defuzzification process is applied to κ (ξ_i) values. The one with the highest score is the best alternative in this step.

5.3 Inferences and discussions

In both of the solution methods -fuzzy EDAS and fuzzy TODIM- the industrial site warehouse alternative is chosen in application of vertical farming in urban area. Although this alternative’s initial cost is the highest in total, comparing it with the others in total would be improper due to its 200 m² area. So, we have calculated all the alternatives’ unit meter square cost. Of course in that case industrial site warehouse alternative has had the lowest cost. In both methods’ steps the initial cost criterion has more weight in the decision making procedure, that’s the reason why this alternative is selected as the best one. The second alternative is the modular one, cause it has more flexibility and least cost. The container alternative is also flexible, however, though two or more containers could be conglomerated, it induces volume constraint. Although the result is surprising, in the market, the stock-out cost is important and to make money you need to sell more in compatible with the product volume criterion. The fuzzy EDAS method ensures the intervals from the averages of the criteria, besides the fuzzy TODIM method procures a solution process for risky problems. In this vertical urban farming application area, the investment

plan is indeed vague and risky. That’s why we utilized and checked the fuzzy EDAS method’s results with the fuzzy TODIM method. And, in addition the fuzzy TODIM method ensures us to employ sensitivity analysis through to be sure about the robustness of the results. As can be seen from the table above, in all other risky options the proper order does not change. Both of the solution methods provide more close results to real life.

Next section exhibits the concluding remarks and future study offerings.

6 Conclusion

Sustainability enforces the people in the society to search feasible solutions for feeding the society. Vertical farming in the city center or edge of a city is one of the rational remedy for the nourishment problem, especially in growing and chaotic metropoles.

In this study we tried to provide various alternatives for the cited problem. To the best of our knowledge, an evaluation process for those alternatives is offered for the first time in the literature. In addition while the topic is hot and full of ambiguities, there are unhealthy data and point of views. Criteria for the evaluation process were determined by the consensus of farmers engaged in vertical farming, and agricultural engineers after the search from the literature. Due to the process’ vagueness, fuzzy MCDM was offered for the selection.

The fuzzy EDAS method conforms the imprecise situations, and due it measures distances from the average, it provides better results according to this problem. Among three alternatives: Container, Modular, and Industrial Site Warehouse; the last one is chosen after applying the fuzzy EDAS Method. Because all the decision makers gave high importance points to the production volume and the stock-out cost criteria, this is reasonable.

Fuzzy TODIM method also provided the Industrial Site Warehouse as the best alternative among all. In addition, other two alternatives did not change their ranks thanks to TODIM’s risk aware calculation process. The sensitivity analysis ensured the robustness of the result.

For the future studies, the net present value technique can be integrated to the solution algorithm. Different MCDM techniques could be applied to the problem. Various improvement optimizations could be offered for each alternative. Location selection for all alternatives separately could be another future study proposition.

Acknowledgment

The authors would like to thank Berkan Gamsiz and Burak Karaaslanoglu for their invaluable contributions to this study.

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